A characterization of translated convex bodies

TL;DR

This paper proves a theorem characterizing when two convex bodies are translates based on their sections by tangent planes of spheres, revealing conditions under which bodies are translated or spherical.

Contribution

It introduces a new characterization of translated convex bodies using tangent sphere sections, extending understanding of convex body symmetries.

Findings

Bodies are translated if tangent sphere sections are translated with equal radii.

If tangent spheres have different radii, the bodies are spheres.

The theorem applies to convex bodies in dimensions three and higher.

Abstract

In this work we present a theorem regarding two convex bodies $K_1, K_2\subset \mathbb{R}^{n}$, $n\geq 3$, and two families of sections of them, given by two families of tangent planes of two spheres $S_i\subset \textrm{int}\textrm{ } K_i$, $i=1,2$ such that, for every pair $\Pi_1$, $\Pi_2$ of parallel supporting planes of $S_1$, $S_2$, respectively, which are corresponding (this means, that the outer normal vectors of the supporting half spaces determined by the two planes have the same direction), the sections $\Pi_1\cap K_1$, $\Pi_2\cap K_2$ are translated, the theorem claims that if $S_1$, $S_2$ have the same radius, the bodies are translated, otherwise, the bodies are also spheres.